Net attacks RAW are made perpetually with disadvantage.
They act as a normal ranged attack, so you are either always throwing them at long range (10-15 feet) and incurring disadvantage, or you are throwing them at short range and as a result are in close combat and throwing them at disadvantage. The description of the net in the PHB gives it no special rules for avoiding either of these two general rules.
Mike Mearls confirms this in a tweet:
Do nets (thrown) use STR or DEX for attack rolls? Are they exempt from close combat disadvantage, as normal range is only 5feet? Dex, since you can't make melee attacks with it. still take disad in close combat despite range. -M
My only thought is the designers worried that nets might be overpowered and that a net attack should always be made at a disadvantage unless the PC is receiving advantage from somewhere. Nets actually seem really underpowered even without the disadvantage problem. Burn an action to restrain a creature who can escape it easily (DC10 or 5 slashing damage), but maybe they worried about the potential for a net dog pile where one player nets an enemy and then everyone makes their attacks against a netted enemy before that enemy gets a chance to escape. In that light it could be broken against single, powerful enemies.
Daggers are melee weapons that have the finesse property. The finesse property allows an attacker to use their dexterity modifier for their attack bonus and damage bonus, in place of the strength modifier.
In this case, the tiefling has a +3 proficiency bonus and a +2 dex modifier, so the attack bonus should be +5 with a finesse weapon, unless there is something missing from the sheet that explains the extra +1 to bring it to +6 such as the dagger being a magic weapon or something. That doesn't appear to be the case here because the equipment for the character is listed on the sheet but something is not adding up correctly, or else I'm missing something on the sheet completely.
As for the piercing damage question, piercing is the type of damage the dagger does. The table in the equipment chapter that details the weapons tells the reader what damage die a weapon uses and also the damage type for the weapon. A dagger does 1d4 damage, and the damage type is piercing. The damage bonus is calculated by adding the attack modifier (in this case dexterity because the weapon has the finesse property) plus any other modifiers to the damage (in this case, none), giving a total of 1d4 + 2.
I'm not certain why the sheet is showing the attack bonus as +6 to hit. All of these sheets seem to have an extra 1 that I can't figure out the source of. If someone finds it, please comment because it's really bothering me and I don't know if I'm just missing something plainly obvious.
I'm leaning towards the calculations on all of these sheets being wrong and chalking it up to an error on the first sheet not being checked and being carried over to every other sheet after. I can see no reason why the bonus is +1 higher than it should be, especially since the spell attack modifier is calculated correctly using the warlock's charisma bonus + proficiency modifier.
Best Answer
With weapons and weapon attack rolls, not having proficiency simply means you do not add the proficiency bonus associated with usual attack rolls. The primary reference to proficiency with respect to attack rolls appears as such in the Player's Handbook:
Note that there is no reference to the user gaining advantage or disadvantage as a result of having or not having proficiency.
You may be confusing this rule with the rule for Armor, where not having proficiency in a given armor type does confer disadvantage. As written in the Player's Handbook:
But like I said, weapon proficiency doesn't affect advantage or disadvantage.